Nick Trefethen, Oxford University

Synopsis

Numerical linear algebra relies on "direct" algorithms, which finish
in finite time, and "iterative" ones, which converge toward the solution but may never
reach it.
Conjugate gradients, introduced in 1952, is the archetypical iterative algorithm, a core
tool of computational science. Yet for the first twenty years of its life, it was generally
regarded as a direct algorithm.
Gaussian elimination, whose roots are in antiquity, is the archetypical direct algorithm.
Yet this algorithm too can be regarded as an iterative one, in which a general matrix is
approximated successively by matrices of rank 1,2,3,.... In recent years this aspect of the
elimination process has become important for applications. This talk will review the
mathematics and the history, with particular attention to the challenging application of
extending Chebfun to higher dimensions, where a continuous analogue of iterative
Gaussian elimination comes into play (joint work with Alex Townsend).